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Biochemical fluctuations, optimisation and the linear noise approximation.

Pahle J, Challenger JD, Mendes P, McKane AJ - BMC Syst Biol (2012)

Bottom Line: Or, which specific changes of parameter values lead to the increase of the correlation between certain chemical species?We implemented our strategy in the software COPASI and show its applicability on two different models of mitogen-activated kinases (MAPK) signalling -- one generic model of extracellular signal-regulated kinases (ERK) and one model of signalling via p38 MAPK.We also investigated correlations between the two parallel signalling branches (MKK3 and MKK6) in this model.

View Article: PubMed Central - HTML - PubMed

Affiliation: School of Computer Science and Manchester Centre for Integrative Systems Biology, The University of Manchester, 131 Princess Street, Manchester, UK. juergen.pahle@manchester.ac.uk

ABSTRACT

Background: Stochastic fluctuations in molecular numbers have been in many cases shown to be crucial for the understanding of biochemical systems. However, the systematic study of these fluctuations is severely hindered by the high computational demand of stochastic simulation algorithms. This is particularly problematic when, as is often the case, some or many model parameters are not well known. Here, we propose a solution to this problem, namely a combination of the linear noise approximation with optimisation methods. The linear noise approximation is used to efficiently estimate the covariances of particle numbers in the system. Combining it with optimisation methods in a closed-loop to find extrema of covariances within a possibly high-dimensional parameter space allows us to answer various questions. Examples are, what is the lowest amplitude of stochastic fluctuations possible within given parameter ranges? Or, which specific changes of parameter values lead to the increase of the correlation between certain chemical species? Unlike stochastic simulation methods, this has no requirement for small numbers of molecules and thus can be applied to cases where stochastic simulation is prohibitive.

Results: We implemented our strategy in the software COPASI and show its applicability on two different models of mitogen-activated kinases (MAPK) signalling -- one generic model of extracellular signal-regulated kinases (ERK) and one model of signalling via p38 MAPK. Using our method we were able to quickly find local maxima of covariances between particle numbers in the ERK model depending on the activities of phospho-MKKK and its corresponding phosphatase. With the p38 MAPK model our method was able to efficiently find conditions under which the coefficient of variation of the output of the signalling system, namely the particle number of Hsp27, could be minimised. We also investigated correlations between the two parallel signalling branches (MKK3 and MKK6) in this model.

Conclusions: Our strategy is a practical method for the efficient investigation of fluctuations in biochemical models even when some or many of the model parameters have not yet been fully characterised.

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Parameter scan of MKKK particle number variance against reaction parameter v2 in the ERK MAPK model. A parameter scan of the variance of the particle number of species MKKK has been carried out for a range of values of the reaction parameter v2, with KI = 45, Vcell = 10-14 l and all other parameters as in [23].
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Figure 3: Parameter scan of MKKK particle number variance against reaction parameter v2 in the ERK MAPK model. A parameter scan of the variance of the particle number of species MKKK has been carried out for a range of values of the reaction parameter v2, with KI = 45, Vcell = 10-14 l and all other parameters as in [23].

Mentions: Presently protein kinases are much better characterised at the molecular level than protein phosphatases. As a consequence the effect of phosphatases are often also not studied in signalling models. However, here we are able to show that the activity of the MKKK-phosphatase does not only influence the type of dynamics the system exhibits, namely that the steady state becomes unstable at v2 = 0.446 due to a Hopf bifurcation. It also strongly affects the intrinsic fluctuations in the system. As can be seen in Figure 3, the estimated variance of MKKK becomes large as v2 approaches the bifurcation point and, interestingly, it shows a local maximum at v2 = 0.32 of 987.7 particles2. The value of v2 in Figure 3 does not go as far as the bifurcation point, as the LNA loses accuracy near this value.


Biochemical fluctuations, optimisation and the linear noise approximation.

Pahle J, Challenger JD, Mendes P, McKane AJ - BMC Syst Biol (2012)

Parameter scan of MKKK particle number variance against reaction parameter v2 in the ERK MAPK model. A parameter scan of the variance of the particle number of species MKKK has been carried out for a range of values of the reaction parameter v2, with KI = 45, Vcell = 10-14 l and all other parameters as in [23].
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3814289&req=5

Figure 3: Parameter scan of MKKK particle number variance against reaction parameter v2 in the ERK MAPK model. A parameter scan of the variance of the particle number of species MKKK has been carried out for a range of values of the reaction parameter v2, with KI = 45, Vcell = 10-14 l and all other parameters as in [23].
Mentions: Presently protein kinases are much better characterised at the molecular level than protein phosphatases. As a consequence the effect of phosphatases are often also not studied in signalling models. However, here we are able to show that the activity of the MKKK-phosphatase does not only influence the type of dynamics the system exhibits, namely that the steady state becomes unstable at v2 = 0.446 due to a Hopf bifurcation. It also strongly affects the intrinsic fluctuations in the system. As can be seen in Figure 3, the estimated variance of MKKK becomes large as v2 approaches the bifurcation point and, interestingly, it shows a local maximum at v2 = 0.32 of 987.7 particles2. The value of v2 in Figure 3 does not go as far as the bifurcation point, as the LNA loses accuracy near this value.

Bottom Line: Or, which specific changes of parameter values lead to the increase of the correlation between certain chemical species?We implemented our strategy in the software COPASI and show its applicability on two different models of mitogen-activated kinases (MAPK) signalling -- one generic model of extracellular signal-regulated kinases (ERK) and one model of signalling via p38 MAPK.We also investigated correlations between the two parallel signalling branches (MKK3 and MKK6) in this model.

View Article: PubMed Central - HTML - PubMed

Affiliation: School of Computer Science and Manchester Centre for Integrative Systems Biology, The University of Manchester, 131 Princess Street, Manchester, UK. juergen.pahle@manchester.ac.uk

ABSTRACT

Background: Stochastic fluctuations in molecular numbers have been in many cases shown to be crucial for the understanding of biochemical systems. However, the systematic study of these fluctuations is severely hindered by the high computational demand of stochastic simulation algorithms. This is particularly problematic when, as is often the case, some or many model parameters are not well known. Here, we propose a solution to this problem, namely a combination of the linear noise approximation with optimisation methods. The linear noise approximation is used to efficiently estimate the covariances of particle numbers in the system. Combining it with optimisation methods in a closed-loop to find extrema of covariances within a possibly high-dimensional parameter space allows us to answer various questions. Examples are, what is the lowest amplitude of stochastic fluctuations possible within given parameter ranges? Or, which specific changes of parameter values lead to the increase of the correlation between certain chemical species? Unlike stochastic simulation methods, this has no requirement for small numbers of molecules and thus can be applied to cases where stochastic simulation is prohibitive.

Results: We implemented our strategy in the software COPASI and show its applicability on two different models of mitogen-activated kinases (MAPK) signalling -- one generic model of extracellular signal-regulated kinases (ERK) and one model of signalling via p38 MAPK. Using our method we were able to quickly find local maxima of covariances between particle numbers in the ERK model depending on the activities of phospho-MKKK and its corresponding phosphatase. With the p38 MAPK model our method was able to efficiently find conditions under which the coefficient of variation of the output of the signalling system, namely the particle number of Hsp27, could be minimised. We also investigated correlations between the two parallel signalling branches (MKK3 and MKK6) in this model.

Conclusions: Our strategy is a practical method for the efficient investigation of fluctuations in biochemical models even when some or many of the model parameters have not yet been fully characterised.

Show MeSH
Related in: MedlinePlus